Primitive Rules Research Articles

First-order logic of change

Abstract We present the first-order logic of change, which is an extension of the propositional logic of change $\textsf {LC}\Box $ developed and axiomatized by Świętorzecka and Czermak (2015, Log. et Anal., 232, 511–527). $\textsf {LC}\Box $ has two primitive operators: ${\mathcal {C}}$ to be read it changes whether and $\Box $ for constant unchangeability. It implements the philosophically grounded idea that with the help of the primary concept of change it is possible to define the concept of time. One of the characteristic axioms for ${\mathcal {C}}$ is ${\mathcal {C}} A \to {\mathcal {C}}\neg A$ and one of the primitive rules is $\omega $-rule introducing $\Box $. It turns out that the next operator is definable in $\textsf {LC}\Box $. Logic $\textsf {LC}\Box $ with next is equivalent to certain fragment of $\textsf {LTL}$ extended by the appropriate definition of ${\mathcal {C}}$. Recently, $\textsf {LC}\Box $ has also been modified to the propositional logic of ‘branching’ changes $\textsf {BTC}$ by Łyczak (2022, Log. J. IGPL) and the logic of ‘parallel’ changes $\textsf {LC}\Box $ ↳ by Świętorzecka and Łyczak (2022, Log. Log. Philios.). Extended in an appropriate manner, $\textsf {BTC}$ is equivalent to a certain fragment of logic $\textsf {CTL}$ to which definitions of two kinds of changes have been added. Here we propose an extension of $\textsf {LC}\Box $ to first-order logic in which, again, the only primitive modal operators are ${\mathcal {C}}$ and $\Box $. We interpret our logic in the semantics of histories of changes. We give an axiomatic system $\textsf {LC}\Box Q$ for the considered logic, and we show selected theses about some relationships between ${\mathcal {C}}$, $\Box $ and $\forall $ (in particular, we prove two versions of the Barcan formula). Then we prove the completeness of $\textsf {LC}\Box Q$. Finally, we compare our logic with $\textsf {FOLT}$ and show the relation between $\textsf {LC}\Box Q$ and a certain fragment of the Kröger system $\varSigma $ to which the definition of ${\mathcal {C}}$ was added.

IPattern

Computational design takes a computer science view of design, applying both the science and art of computational approaches and methodologies to design problems. This article proposes to convert design methodologies studied by designers into rule-based computational design software and help them by providing suggestions for designs to build upon given a set of primitive shapes and geometrical rules. iPattern is a pattern-making software dedicated to designers to generate innovative design patterns that can be used in a decorative manner. They may be applied on wallpapers, carpets, fabric textiles, three-dimensional lanterns, tableware, etc. The purpose is to create a modern pattern design collection that adds a new essence to the place. In order to generate creative design patterns, primitive shapes and geometrical rules are used. The generated design pattern is constructed based on the grid of the Flower of Life of the sacred geometry or similar grids constructed using primitive shapes (rectangles, squares and triangles) combined in the layout of the Flower of Life.

Humanoid Robot: Behaviour Synchronization and Depth Estimation

This work involves various robot behaviours that are required to play soccer using simple primitive rules and then these individual behaviours will be synchronized together so that the humanoid robot is able to play soccer autonomously. The behaviours created are separated into two categories namely object detection behaviours and motion behaviours. In object detection behaviours consists of Red Ball Detection which uses the primitive rules of colour blob segmentation and depth estimation through trigonometric properties. In motion behaviours consists of Scoring, Kip Up and Diving which uses the primitive rule of support polygon and centre of mass. Once these behaviours are designed and created, it is synchronized into two different roles namely player and keeper. The final part of this project is to analyse some of the primitive rules used to design the behaviours so that comparative studies could be done. This project ends with the objectives being achieved and an autonomous humanoid robot which is able to play soccer is successfully being created.

On Hilbert's Axiomatics of Propositional Logic

In this paper I will consider the axioms for propositional logic which were presented by Hilbert in his conferences during the year 1922, and those which were presented by Hilbert and Bernays in the book Grundlagen der Mathematik, I (1934). I will describe a general procedure in order to translate Hilbert's axioms into rules on sequents and I will show that, following this procedure, Hilbert's axioms become particular cases of (derived or primitive) rules of Gentzen's Sequent Calculus and contain ideas which will be focused and developed in Gentzen's Sequent Calculus and also in more recent logical investigations.

The λ-calculus with constructors: Syntax, confluence and separation

AbstractWe present an extension of the λ(η)-calculus with a case construct that propagates through functions like a head linear substitution, and show that this construction permits to recover the expressiveness of ML-style pattern matching. We then prove that this system enjoys the Church–Rosser property using a semi-automatic ‘divide and conquer’ technique by which we determine all the pairs of commuting subsystems of the formalism (considering all the possible combinations of the nine primitive reduction rules). Finally, we prove a separation theorem similar to Böhm's theorem for the whole formalism.

The Invention of Legal Primitivism

This Article addresses a different sort of legal transplant — one in which outside legal doctrines are imported in order to be cabined, treated as normative counterpoints, and identified as the legal other. Legal primitivism is a kind of anti-transplant. It heightens the persistent differences between a dominant legal system and its understanding of primitive rules. An often ignored legal literature depicting legal primitivism emerged in the second half of the nineteenth century and in the early twentieth century. Mapping the differences between America’s modern legal system and its antecedents, this immense literature, which included works by OliverWendell Holmes, James Coolidge Carter, and John Henry Wigmore, described an archaic legalism which sometimes belonged to tribal societies, and sometimes was simply conjured out of thin air. Exploring the project of constructing geographies of legal knowledge assists in our understanding of American law in a period of significant change. What elements of primitive law were valorized? Which were seen as archaic or repugnant? And what was the purpose of constructing a legal doppelganger? By examining these cultural negotiations, and holding legal primitivism up as a mirror to modern law, it is possible to uncover the anxieties of legal modernism.

Creating operational shift schedules for third-level IT support: challenges, models and case study

IT support can be divided into first-level support, second-level support and third-level support. Although there is a large body of existing work regarding demand forecasting and shift schedule creation for various domains such as call centres, very little work exists for second- and third-level IT support. Moreover, there is a significant difference between such support and other types of services. As a result, current best practices for scheduling such work are not based on demand, but rather on primitive rules of thumb. Due to the increasing number of people providing such support, theory and practice is sorely needed for scheduling second- and third-level support shifts according to actual demand. In this work, we present an end-to-end methodology for forecasting and scheduling this type of work. We also present a case study in which this methodology demonstrated significant potential savings in terms of manpower resources.

When US Medicine Became Imperial

At least, it is a pity that author of JAMA's 1897 Mis¬ cellany on Jose Rizal, MD, PhD, did not say more about Rizal and late 19th-century situation ofthePhilippinesunderSpan¬ ish rule. Although article mentions Rizal's medical training in Spain, Germany, and Paris, author focuses on exotic aspects of story: odd beliefs and practices of the Malay race, their bloody rituals, and Rizal's snake-like expression and repulsive face, immobile, yellow, passive. Instead of char¬ acterizing Rizal and other Filipinos in egregiously racist terms, author might have considered nationalist aspirations ofa people subjected to an increasing economic burden imposed by their Spanish masters. The Spanish Revolution of 1868 and opening of direct steamship travel betweenManila andMadrid through Suez Canal in 1869 led to establishment of a Spanish-directed spoils system in Philippines that transferred increasing amounts of land and wealth to Spanish colonists and corrupt locals. Rizal provided a balanced account of some of these changes in his 1886 novel, Nolime tangere. His reform efforts of 1880s in Young Filipino Party aimed to reduce influence of both a corrupt foreign clergy and a village-based native elite. An account ofRizal's fate may have been included in an 1897 JAMA only because Rizal was a physician. None¬ theless, just as Philippine physicians had become involved in Spanish imperial politics, US victory in Spanish-Ameri¬ can War of 1898 soon involved US physicians in Philip¬ pines, Cuba, and Puerto Rico. United States acquisition ofa tropical island empire brought with it a long-recognized challenge: vulnerability of Euro¬ pean races to tropical climates, especially to yellow fever and malaria. To overcome this problem, federalmilitary and health officials considered tropical medicine and hygiene as essential components of a successful imperial program, a necessary pre¬ condition for sustaining US political, military, and economic in¬ terests. Even before US invasion of Cuba in 1898, public health officials had been concerned with risks that Cuba's endemic fevers posed to continental United States. Cuba, so close to Florida and neglected by Spanish health authorities, was seen as a threatening source of infections, especially lep¬ rosy and yellow fever. Indeed, in 1896 one concerned US phy¬ sician urged invasion of Cuba on grounds of removing a threat to mainland public health: view of necessity for protection ofourpeople from thispestilence... is it not duty of United States Government to intervene, by force if need be, and to afford security to Americans from a danger, ef¬ fects ofwhich are more horrible than ravages ofwar?1 Convinced of power of rational and authoritative public health management, hygienic officials of US Army quickly asserted their control over newly acquired tropical places and populations. For example, when cholera struck Philip¬ pines in 1902, its new US-controlled Board of Health immedi¬ ately responded with strict measures of isolation and disinfec¬ tion. According to US officials, only limit to their successwas noncompliance by many Filipinos, whom one derided for their purported fatalism, superstition, filthy habits, and univer¬ sal disregard of most primitive rules of hygiene.2 Officers who observed hygienic measures remained largely untouchedby cholera; enlistedmenand especially indigenous people suffered more, confirming elite perceptions of limited ca¬ pacity ofboth Filipinos and poorly educated soldiers.2 In Cuba,

Lazy techniques for fully expansive theorem proving

The HOL system is afully expansive theorem prover: proofs generated in the system are composed of applications of the primitive inference rules of the underlying logic. One can have a high degree of confidence that such systems are sound, but they are far slower than theorem provers that exploit metatheoretic or derived properties.

A theory of binding structures and applications to rewriting

In this paper we present a theory of binding structures, and give examples of its application to rewriting. We define the set of binding structures as an abstract algebra, and define a general notion of parameterized homomorphism. A variety of operations on binding structures are presented as homomorphisms, and a collection of properties useful for developing applications is given. Three applications are presented: a generalized notion of term rewriting: a theory of unification for binding structures; and a set of structures and primitive rules intended to serve as a basis for design of rewriting components for (quantified) first-order reasoning systems.

Real reduced models for relevant logics without ${\rm WI}$.

Slaney has provided reduced models (ones in which there is but one world) for a number of relevant logics via certain kinds of frames, as opposed to the conventional Routley-Meyer model structures. This paper does three things: it corrects Slaney's paper, extends his results in a different direction, and draws a moral from the errors it corrects. The corrections to Slaney's paper are very minor, the errors having been more in the nature of slips than of outright mistakes. The semantic structures of Slaney's paper are criticized for not being enough. It is then shown that Slaney's basic results can be used to provide reduced models for most of the same logics (the system E being a notable exception) using the Routley-Meyer model structures which do not suffer from this defect. The basic slip in the original paper was not to close the worlds of the canonical models of some of the systems under all of the primitive rules of inference of that system. The paper ends with a brief discussion of the philosophical significance of insisting that theories (worlds) be closed under certain rules of inference as well as under provable implication. That discussion insists upon the importance of a distinction between primitive/derivable rules of inference and merely admissible rules along the lines of Anderson and Belnap. / Introduction Slaney [8] discusses the motivationally important matter of reduced modeling for relevant logics, duly notes that many important weaker relevant logics have not been provided with reduced modeling, and goes on to offer such for them in terms of frames, as opposed to the conventional model structures of Routley and Meyer [5],[6] for instance. In addition to their motivational importance, reduced models are technically and practically important for the practicing logician. They are simpler and hence easier to use. However we find [8] lacking in some very important respects. In the first place, there are some minor (i.e., easily fixed) technical inaccuracies in the paper. Some of the claims made therein are false as stated, and some Received June 24, 1991; revised September 12, 1991 REDUCED MODELS FOR RELEVANT LOGICS 443 of the proffered proofs are unsound. So in what follows we will revise the claims of [8] as necessary and at least indicate how the incorrect proofs can be made right. (It is very unlikely that the author of [8] was actually mistaken on any of these points. The errors corrected herein were surely just slips.) But our discontent with [8] runs deeper: the models of [8] are not in general really models —in a very clear sense, they are not semantical enough! As the reader will see when we get around to introducing the technical definitions, for a given logic an L-frame is defined in terms of the crucial syntactic notions of 'theorem of L' and 'primitive rule of L Now there may be those who are not purists in these matters and whose sensibilities are not offended by such wanton self-indulgence. We ourselves are very open-minded on this point and would not care to judge them to be moral reprobates. However, it is our duty to warn them that they will pay for their sins immediately in this case. For this illicit conjoining of syntax and semantics begets here a degenerate offspring: frames so defined are useless for many of the tasks to which logicians are wont to appoint them, e.g., proving that a given formula (neither already known to be a theorem nor known to be a nontheorem of L) is valid or invalid as the case may be. It is not seriously being suggested that the results of [8] are unworthy. The extension of metavaluation techniques given there is particularly admirable. But such upright works should be put at the service of holier tasks. So in addition to correcting [8], we will extend it. In particular, we prove that the basic results of [8] can be used to show that almost all of the logics treated there are characterized by real reduced models, i.e., ones based on the traditional Routley-Meyer relational model structures. So without further ado, the technical details. 2 Syntactic and semantic preliminaries The systems to be considered can be fitted with Hilbert-style axiomatizations from the following list of axioms and rules (the side notation designates the corresponding semantic postulate as listed further below): (Al) A->A pi (A2) A&B^A (A3) A&B-+B (A4) (A -* B) & (A -> C) -* .A -* B & C p2.i (A5) A^AvB (A6) B-+AvB (A7) (A^C)&(B-+C)-+.AvB-+C p2.i (A8) A&(BvC)-+(A&B)vC (A9) ^A-*A p4 (A10) A^-^B-».B-*-yA p7 (All) A-+B-+ .C-+A-+ .C->£ pό.i (A12) A-+B->.B-+C->.A-+C pό.ii (A13) (A-+B) & (B^C)-+.A->C p9 (A14) A->.A->A pl2 (A15) A^.B-+A pl3 (A16) (A-+BvC)& (A&B^O^.A^C ? (A17) A-+ .A->B-+B plO (A18) (A-+ .B-+C)-+ .£-• .A-+C pll

Rule-guided behavior and self-monitoring on the tower of hanoi disk-transfer task

Problem-solving efficiency, rule-based strategies, and self-monitoring were observed on a novel planning task that provided feedback, practice, and the opportunity for strategy revision. The 3- and 4-Disk Tower of Hanoi (TOH) task was administered to 100 children ranging in age from 3 to 12 years, depending upon age and success on the 3-Disk version. On the 3-Disk task, efficiency improved with age and deteriorated with more complex problems; age differences increased with problem complexity. Efficiency on the 4-Disk task was affected by problem complexity only. Error types were surprisingly similar across the age range, reflecting primitive rules that reemerged in performance when problems exceeded processing capacity. The self-monitoring score on the 3-Disk task showed age changes and predicted planning efficiency only moderately. The particular task environment and performance analysis used here revealed both better performance in younger children than had been found previously, and “regressions” in the performance of older children in response to reaching their problem solving “limit.”

Numerical mathematics of Feynman path integrals and the operator ordering problem.

Infinitesimal propagators have been used in quantum-mechanical problems to establish ordering rules for classical quantities. Ambiguities arise in operator ordering as a result of the freedom of choice in evaluating the classical action over a short-time interval. The primitive quadrature rule employed in a short-time propagator establishes the quantum system associated with the classical Hamiltonian. Normal propagators are constructed from products of infinitesimal propagators and represent a compound quadrature in which the path is broken into line segments and a separate quadrature rule is applied to each line segment. It is demonstrated for Hamiltonians involving the vector potential that higher-order quadratures of the infinitesimal propagator all give a unique and proper ordering. Similarly, Hamiltonians that are products of powers of position x and momentum p, i.e., ${\mathit{p}}^{\mathit{l}}$${\mathit{x}}^{\mathit{m}}$, show some preference for Born-Jordon ordering as higher quadrature rules are applied. Unique orderings can be obtained in some cases by solving for the normal propagator. This is done for the Hamiltonian ${\mathit{p}}^{\mathit{l}}$${\mathit{x}}^{\mathit{m}}$ by using a Fourier representation for the phase-space propagator. This formalism requires that both the momentum and position coordinates be represented by Fourier series. These results show that care must be taken when generating quantum-mechanical operators by using primitive quadrature rules in conjunction with a short-time propagator. These ambiguities are eliminated when the compound quadrature is performed to generate the normal propagator.

Formalisations of further ℵ0-valued Łukasiewicz propositional calculi

It has been shown that, for all rational numbers r such that 0≤ r ≤ 1, the ℵ0-valued Łukasiewicz propositional calculus whose designated truth-values are those truth-values x such that r ≤ x ≤ 1 may be formalised completely by means of finitely many axiom schemes and primitive rules of procedure. We shall consider now the case where r is rational, 0≥r≤1 and the designated truth-values are those truth-values x such that r≤x≤1.We note that, in the subcase of the previous case where r = 1, a complete formalisation is given by the following four axiom schemes together with the rule of modus ponens (with respect to C),the functor A being defined in the usual way. The functors B, K, L will also be considered to be defined in the usual way. Let us consider now the functor Dαβ such that if P, Dαβ take the truth-values x, dαβ(x) respectively, α, β are relatively prime integers and r = α/β thenIt follows at once from a theorem of McNaughton that the functor Dαβ is definable in terms of C and N in an effective way. If r = 0 we make the definitionWe note first that if x ≤ α/β then dαβ(x)≤(β + 1)α/β − α = α/β. HenceLet us now define the functions dnαβ(x) (n = 0,1,…) bySinceit follows easily thatand thatThus, if x is designated, x − α/β > 0 and, if n > − log(x − α/β)/log(β + 1), then (β + 1)n(x−α/β) > 1.

On the structural completeness of some pure implicational propositional calculi

1. The notion of the structural completeness of systems was introduced by W. A. Pogorzelski in [4]. The main result of this article is a theorem on structural completeness of Hubert's implicational propositional calculus. We also prove that Lewis's implicational cal? culi1 S5 and S3 are not structurally complete. 2. Let piqis,p1,p2, ... be an infinite sequence of various propositional variables. Let S stand for the least of the sets containing all propositional variables and closed with regard to connective ->. The variables ol, ?, y represent elements of the set S, and A, X, Y those of the set 2s. By At we denote the set of all propositional variables, by Ve the extension of the function e, transforming the set At into S, to the endo morphism of the algebra , and thus: Ve(p) = e (p), Ve(tx -> ?) = Vc(ol) -> -> Fe(?), for any a, ? e S, p e At. Moreover, we assume the symbols and definitions employed by W. A. Pogorzelski in [4], Let us now repeat Pogorzelski's definition: A system with primitive rules R and axioms A is structurally complete (in symbolic notation: e rnx = S = n A A (.Ve(n) F?(a) e X) , e: At -> 5

A Pair of Primitive Rules for the Sentential Calculus

A Pair of Primitive Rules for the Sentential Calculus

On the scope of the classical deduction theorem

§1. The set Cn(R, A + X) is the set of all expressions deducible, on the ground of formulas A, from the class X of formulas of the implicational propositional calculus, by means of rules belonging to the set R. In other words, Cn(R, A + X) is the set of expressions obtained from the set X by the help of a two-parameter function Cn[R, A]. The formula is called the classical deduction theorem.The classical deduction theorem is true for the system 〈R, A〉 (where R is the set of primitive rules, A is the set of axioms of the propositional calculus) if it holds for the function Cn[R, A].

The adequacy of the theories of deductive systems with respect to sentential calculi

The sentential calculiR, under discussion, are axiomatizable and implication is among their primitive terms. The modus ponens and the rule of substitution are their primitive rules. ByS r is denoted the set of sentences obtained from the formulae of the calculusR by substituting sentences of a given language for all variables. The variablesx, y, z ... represent the elements of the setS r , the variablesX, Y, Z ... represent the subsets ofS R . The formulacxy designates an implication withx as its antecedent andy as its consequent,cxy is always an element ofS R δ(X) means, thatX is closed with respect to the modus ponens rule.A R designates the class of allS R -substitutions of the axioms of the sentential calculusR.

A definition of existence in terms of abstraction and disjunction

Greater economy can be effected in the primitive rules for the system K of basic logic by defining the existence operator ‘E’ in terms of two-place abstraction and the disjunction operator ‘V’. This amounts to defining ‘E’ in terms of ‘ε’, ‘έ’, ‘o, ‘ό’, ‘W’ and ‘V’, since the first five of these six operators are used for defining two-place abstraction.We assume that the class Y of atomic U-expressions has only a single member ‘σ’. Similar methods can be used if Y had some other finite number of members, or even an infinite number of members provided that they are ordered into a sequence by a recursive relation represented in K. In order to define ‘E’ we begin by defining an operator ‘D’ such thatHere ‘a’ may be thought of as an existence operator that provides existence quantification over some finite class of entities denoted by a class A of U-expressions. In other words, suppose that ‘a’ is such that ‘ab’ is in K if and only if, for some ‘e’ in A, ‘be’ is in K. Then ‘Dab’ is in K if and only if, for some ‘e and ‘f’ in A, ‘be’ or ‘b(ef)’ is in K; and ‘a’, ‘Da’, ‘D(Da)’, and so on, can be regarded as existence operators that provide for existence quantification over successively wider and wider finite classes. In particular, if ‘a’ is ‘εσ’, then A would be the class Y having ‘σ’ as its only member, and we can define the unrestricted existence operator ‘E’ in such a way that ‘Eb’ is in K if and only if some one of ‘εσb’, ‘D(εσ)b’, ‘D(D(εσ))b’, and so on, is in K.

Banishing the rule of substitution for functional variables

Let and be (well-formed) formulas of the functional calculus, let c be an n-adic functional variable, and let a1, …, an be distinct individual variables. Church has defined the metalogical notation to indicate the formula resulting from when each part of of the form c{1, …, n) (such that the occurrence of c is free in ) is replaced by the formula which arises from by replacing every free occurrence of ai by i, i, = 1, …, n. (Here 1, …, n may be any individual variables or constants, not necessarily all distinct.) The notation is not defined for all , , c, a1, …, an, however, but only for those cases (specified in detail by Church) when the resulting formula constitutes a valid substitution instance of the formula according to the standard interpretation of the functional calculus.The full and correct syntactical statement of the conditions (under which this type of substitution is permissible) has proved so arduous, that it seems to have been rendered in error more often than not. Unfortunately the functional calculi are often set up as deductive systems in which this type of substitution occurs as one of the primitive rules of inference, or in one of the axiom schemata. Thus the beginning student who is introduced to the calculi through such a formulation is forced to cope from the outset with details which have proved treacherous even to the initiate. For this reason it is desirable to seek alternative formulations of the functional calculi in which this type of substitution is not mentioned.